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Conics: General Form

The general equation of a second-degree curve is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.

Parabola: $A=0, B=0, C=1, D=-4a, E=0, F=0$ (e.g., $y^2 = 4ax$)
Ellipse: $A \neq C$, $B=0, D=0, E=0, F=-1$ (e.g., $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$)
Hyperbola: $A=-C$, $B=0, D=0, E=0, F=-1$ (e.g., $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$)

Eccentricity ($e$)

Definition: $e = \frac{PS}{PM}$ (ratio of distance from focus to distance from directrix).
Parabola: $e=1$
Ellipse: $0 < e < 1$ ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $e^2 = 1 - \frac{b^2}{a^2}$ if $a>b$)
Hyperbola: $e > 1$ ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, $e^2 = 1 + \frac{b^2}{a^2}$)

Parabola ($y^2 = 4ax$)

Focus: $(a, 0)$
Directrix: $x = -a$
Eccentricity: $e=1$

Ellipse ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$)

Foci: $(\pm ae, 0)$
Directrices: $x = \pm \frac{a}{e}$
Eccentricity: $e = \sqrt{1 - \frac{b^2}{a^2}}$ (for $a>b$)
Sum of distances from foci: $S'P + SP = 2a$

Hyperbola ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$)

Foci: $(\pm ae, 0)$
Directrices: $x = \pm \frac{a}{e}$
Eccentricity: $e = \sqrt{1 + \frac{b^2}{a^2}}$

Rotation of Axes

If the coordinate axes are rotated by an angle $\alpha$, the new coordinates $(x', y')$ are related to the old coordinates $(x, y)$ by:

$x = x' \cos\alpha - y' \sin\alpha$
$y = x' \sin\alpha + y' \cos\alpha$

The general equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ transforms to $A'x'^2 + B'x'y' + C'y'^2 + D'x' + E'y' + F' = 0$.

To eliminate the $xy$ term ($B' = 0$), rotate by $\alpha$ such that $\tan(2\alpha) = \frac{B}{A-C}$.

The discriminant $B^2 - 4AC$ is invariant under rotation: $B'^2 - 4A'C' = B^2 - 4AC$.

Classification of Conics by Discriminant

For $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$:

If $B^2 - 4AC = 0$: Parabola
If $B^2 - 4AC < 0$: Ellipse (or Circle if $A=C$ and $B=0$)
If $B^2 - 4AC > 0$: Hyperbola

Parametric Forms

Parabola ($y^2 = 4ax$)

$x = at^2$
$y = 2at$

Ellipse ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$)

$x = a \cos\theta$
$y = b \sin\theta$

3D Geometry: Direction Cosines and Ratios

Direction Cosines (d.c.'s)

If a line makes angles $\alpha, \beta, \gamma$ with the positive $x, y, z$ axes, its direction cosines are $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$.

Property: $l^2 + m^2 + n^2 = 1$

Direction Ratios (d.r.'s)

Any set of numbers $a, b, c$ proportional to the direction cosines $l, m, n$ are called direction ratios. So, $\frac{l}{a} = \frac{m}{b} = \frac{n}{c} = k$.

If $(a, b, c)$ are d.r.'s, then d.c.'s are: $l = \frac{a}{\sqrt{a^2+b^2+c^2}}$, $m = \frac{b}{\sqrt{a^2+b^2+c^2}}$, $n = \frac{c}{\sqrt{a^2+b^2+c^2}}$
A line passing through points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ has d.r.'s $(x_2-x_1, y_2-y_1, z_2-z_1)$.

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